| L(s) = 1 | − 3-s + 5-s − 2·7-s − 2·9-s − 4·11-s + 13-s − 15-s − 4·19-s + 2·21-s − 23-s + 25-s + 5·27-s − 7·29-s − 7·31-s + 4·33-s − 2·35-s − 4·37-s − 39-s + 3·41-s + 6·43-s − 2·45-s − 13·47-s − 3·49-s + 10·53-s − 4·55-s + 4·57-s − 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s − 2/3·9-s − 1.20·11-s + 0.277·13-s − 0.258·15-s − 0.917·19-s + 0.436·21-s − 0.208·23-s + 1/5·25-s + 0.962·27-s − 1.29·29-s − 1.25·31-s + 0.696·33-s − 0.338·35-s − 0.657·37-s − 0.160·39-s + 0.468·41-s + 0.914·43-s − 0.298·45-s − 1.89·47-s − 3/7·49-s + 1.37·53-s − 0.539·55-s + 0.529·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.74228153273648058530396531676, −9.807550538439381143997160199045, −8.887175936832334481566782750977, −7.87386866029284703957997986543, −6.68033008537858468908242123188, −5.83627871217972681785142914498, −5.11452995004582337384367234955, −3.55327009794682067150063225289, −2.28205647017346971309572600196, 0,
2.28205647017346971309572600196, 3.55327009794682067150063225289, 5.11452995004582337384367234955, 5.83627871217972681785142914498, 6.68033008537858468908242123188, 7.87386866029284703957997986543, 8.887175936832334481566782750977, 9.807550538439381143997160199045, 10.74228153273648058530396531676
